Asymmetric simple exclusion process: Difference between revisions

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{{short description|Interacting particle system}}

{{one source|date=December 2012}}

In [[probability theory]], the '''asymmetric simple exclusion process (ASEP)''' is an [[interacting particle system]] introduced in 1970 by [[Frank Spitzer]].<ref>{{Cite injournal
| ''last=Spitzer | first=Frank |author-link = Frank Spitzer
| year=1970
| title=Interaction of Markov Processes''
| journal=[[Advances in Mathematics]]
| volume=5 | issue=2 | pages=246–290
| doi=10.1016/0001-8708(70)90034-4 | doi-access=free}}</ref> Many articles have been published on it in the physics and mathematics literature since then, and it has become a "default stochastic model for transport phenomena".<ref>{{cite journal|last=Yau|first=H.T.|author-link=Horng-Tzer Yau

|title=(log t)^2/3 law of the two dimensional asymmetric simple exclusion process|journal=Ann. Math. |year=2004|volume=159|pages=377–405|doi=10.4007/annals.2004.159.377|arxiv=math-ph/0201057|s2cid=6691714}}</ref>

The process with parameters <math>p, q \geqslant 0,\, p + q = 1</math> is a continuous-time [[Markov process]] on <math>S = \lbrace 0, 1\rbrace^{\mathbb{Z}}</math>,

the 1s being thought of as particles and the 0s as holesempty sites. Each particle waits a random amount of time having the distribution of an exponential random variable with mean one and then attempts a jump, one site to the right

exponent mean one amount of time and then attempts a jump, one site to the right

with probability <math>p</math> and one site to the left with probability <math>q</math>. However, the jump is performed only if there is no particle at the target site. Otherwise, nothing happens and the particle waits another exponential time. All particles are doing this independently of each other.

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==References==

*{{Citation

| last=Spitzer | first=Frank |author-link = Frank Spitzer

| year=1970

| title=Interaction of Markov Processes

| journal=[[Advances in Mathematics]]

| volume=5 | issue=2 | pages=246–290

| doi=10.1016/0001-8708(70)90034-4}}.

*{{Citation

| last1=Tracy | first1=C. A. | author1-link = Craig Tracy

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| volume=290 |issue=1 | pages=129–154

| doi=10.1007/s00220-009-0761-0

|arxiv = 0807.1713 |bibcode = 2009CMaPh.290..129T | s2cid=14730756 }}.

*{{Citation

| last1=Bertini | first1=L.

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| journal=[[Communications in Mathematical Physics]]

| volume=183 |issue=3 | pages=571–607 | doi=10.1007/s002200050044|bibcode = 1997CMaPh.183..571B | citeseerx=10.1.1.49.4105

| s2cid=122139894

}}.